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Space-Time Foliation in Quantum Gravity

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

The aim of this chapter is to provide all basic information necessary to read the thesis and to clarify our stance on discussing quantum gravity. Through Sect. 1.1, one can understand why we think that an object called space-time foliation is useful in quantum gravity. The book treats two theories of gravity with the space-time foliation: Causal Dynamical Triangulations (CDT) and n-DBI gravity; the following sections are devoted to explaining their fundamental ideas and to supplements for understanding the theoretical frameworks. Namely, since CDT is a lattice formulation of quantum gravity, in Sect. we show how to put gravity on a lattice with and without the space-time foliation; Sect. 1.3 leads readers to n-DBI gravity starting with the idea of the 3 + 1 decomposition which is a standard way to investigate the gravitational physics with the space-time filiation.

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Notes

  1. 1.

    As for the notation of constraints, see Sect. 1.3.

  2. 2.

    See, for instance, [9].

  3. 3.

    There is a nice review of CDT [9]; in this book, we follow the notation appeared there.

  4. 4.

    In \(2\) dimensions the difference of \(\alpha \) can be absorbed into the redefinition of the cosmological constant, so that one can set \(\alpha =1\) without loss of generality.

  5. 5.

    For a good review of the Liouville field theory, see [33].

  6. 6.

    The subscripts of constraints may seem to be weird, say why does not \(\varPhi _{3}\) exist? This convention has been made for the later convenience.

  7. 7.

    A well-known example is a chiral boson.

  8. 8.

    The first two issues are apparently contradictory: if the lapse must collapse everywhere, there would be no way to develop an exponentially growing mode involving the lapse. We will, however, discuss the instabilities of the \(n\)-DBI model which evades the first issue.

  9. 9.

    We thank Soo-Jong Rey and Takao Suyama for suggesting this formulation.

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  1. Department of Physics, Nagoya University, Nagoya, Japan

    Yuki Sato

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Sato, Y. (2014). Introduction. In: Space-Time Foliation in Quantum Gravity. Springer Theses. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54947-5_1

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